Why Is the tt Component of the Metric Interpreted Differently in This Paper?

[vizart]

I am reading a paper and the authors read the tt component of the metric from the line element $$ds^2=f(r)[g(r) dt^2+h(r) dr^2]$$ as $$g_{tt}=g(r)$$ instead of (what I expect to be) $$g_{tt}=f(r) g(r)$$

Could somebody please explain to me why? Thank you.

 

[George Jones]

This seems strange. What is the paper?